OFFSET
0,4
COMMENTS
A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{k=0..n-1} k*A143953(n,k).
G.f.: z^2*C/[(1-z)^2*sqrt(1-4z)], where C = [1-sqrt(1-4z)]/(2z) is the Catalan function.
a(n) ~ 2^(2*n+1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: (-n+1)*a(n) +2*(3*n-4)*a(n-1) +(-9*n+13)*a(n-2) +2*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jun 16 2016
EXAMPLE
a(3)=5 because in the peak plateaux of the Dyck paths UDUDUD, UD(UUDD), (UUDD)UD, (UUDUDD) and U(UUDD)D, shown between parentheses, we have 0 + 1 + 1 + 2 + 1 = 5 peaks.
MAPLE
C:=((1-sqrt(1-4*z))*1/2)/z: G:=z^2*C/((1-z)^2*sqrt(1-4*z)): Gser:=series(G, z= 0, 30): seq(coeff(Gser, z, n), n=0..25);
MATHEMATICA
CoefficientList[Series[x^2*((1-Sqrt[1-4*x])*1/2)/x/((1-x)^2*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) x='x+O('x^50); concat([0, 0], Vec(x*(1-sqrt(1-4*x))/(2*(1-x)^2*sqrt(1-4*x)))) \\ G. C. Greubel, Mar 22 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 10 2008
STATUS
approved